determine a $3N-K$ dimensional "surface" in the $3N$ dimensional abstract Euclidean space $E^{3N}$ in which one point specifies
collectively the position of all the $N$ particles. Call this surface $Q$. Note that this a geometric object independent of coordinate
systems. This is the "configuration manifold". Now consider all the $N$ vectors $\tau_i$ each with three components as one grand
vector in the large space $E^{3N}$ in which the configuration manifold rests. Then this grand vector is tangent to the configuration manifold.
This is how the tangency condition which is apparent with one constraint generalizes. To see that this is the case note
that the grand $\tau$ vector with $3N$ components is tangent to the surface $f_1=0$. This follows from Eq.(2) above
as $\nabla_i f_1$ is a normal to this surface. Similarly the grand vector is tangent to $f_2=0$ etc.. Since the intersections of these
surfaces is the configuration manifold, it follows that the grand vector is tangent to $Q$.

Generalized Coordinates

Now we choose coordinates on the configuration space or manifold. This would make the constraint equations trivial.
Say we have new coordinate system $\{q_1,q_2,\cdots,q_{3N}\}$ with the following transformation equations:

Pick the $q$ such that that the constraint equations reduce to a condition of constancy of some of them.
Its clear that exactly $K$ such trivial coordinates exist, say these are the last $K$ of the $3N$ generalized coordinates.
Thus we write these are functions of the constraint equations:

where $n=3N-K$.
These can be inverted to wirte the constraints purely in terms of the last $K$ generalized coordinates.

(7)

\begin{align} f_k(\mathbf{r})=f_k(q_{n+1},\cdots,q_{3N}) \end{align}

When the constraints are imposed this implies that $q_{n+k}=R_k(0,0,\ldots,0)$ which are clearly constants.

Thus there are only $n$ non-trivial generalized coordinates and these form a coordinate system on the
configuration manifold. The number $n$ is the number of degrees of freedom often simply d.o.f
and is a crucial number. For example $n=1$$ systems are totally different from general $n=2$ systems
which are themselves quite different from $n=3$ systems. Of course for a gas which has an avogadro number of
particles the number of d.o.f is so large that adding or deleting one is not going to make a difference. On the other hand
interesting statistical properties arise for such large numbers that leads to thermodynamics.

Problems

What is the configuration space of a spherical pendulum?

Same for a double spherical pendulum

Same for a planar double pendulum

Earth (tricky!). Your arms (more tricky?)

For a particle in the inside surface of a vertical inverted cone write the three generalized coordinates in term of cartesian coordinates that are such that they have the desired properties above? What are the functions $R_k$ in this case?